PHYS 4392 Electromagnetism
T.E. Coan
~ × (B
~ × C)
~ = B(
~ A
~ · C)
~ − C(
~ A
~ · B)
~
Proof of vector identity A
~ × (B
~ × C)
~ = B(
~ A
~ · C)
~ − C(
~ A
~ · B)
~ found on the inside of
We seek to prove the vector identity A
the cover of Griffiths and used often in electromagnetism. This is sometimes called the “bac - cab”
(pronounced “back minus cab”) rule. Using our definition of the cross product from lecture,
~ × (B
~ × C)]
~ i
[A
~ × C)
~ k
= ǫikj Aj (B
= ǫijk Aj ǫklm Bl Cm ,
(1)
(2)
where ǫikj is the Levi-Civita tensor. Note that since I already used the suffixes j and k in the first
cross product, I need to use something different for the second cross product. I chose l and m. You
could have chosen, say, r and s.
A useful identity we will not prove that involves a pair of Levi-Civita tensors is
ǫijk ǫklm = δil δjm − δim δjl
(3)
where,
δij =
1 if i = j
0 if i 6= j
is the Kronecker delta you recall from childhood. Note the order of the indices in the boxed equation.
There is a pattern.
Rewriting the RHS of (2) and using (3) yields
Aj δil Bl δjm Cm − Aj δjl Bl δim Cm = Aj Bi Cj − Aj Bj Ci
= Bi Aj Cj − Ci Aj Bj
(4)
(5)
Recall that we can shuffle factors like Aj or Cm without thinking since they are just numbers.
Shuffling the indices of a factor is, in general, a very different story. Examining (5) shows it is
~ A
~ · C)
~ − C(
~ A
~ · B).
~ Ergo,
nothing more than B(
~ × (B
~ × C)
~ = B(
~ A
~ · C)
~ − C(
~ A
~ · B)
~
A
1
(6)